Subsurface geological modeling involves predicting target variables of interest for development planning and production forecasting. Target variables are directly measured at locations where the subsurface has been penetrated by wells through which various tools are run to take measurements. The subsurface model at the sample locations is conditioned by these sample measurements with no attached uncertainty. Between well locations, however, predictions of target variables with attached measures of uncertainty are required.
To reduce prediction uncertainty, a wide variety of subsurface attributes ranging in scope, scale, and quality can be considered as predictive proxies of the target variable where direct measurements do not exist. Categories and examples of attributes might include geometric (formation thickness, formation depth, fault distance, fracture intensity), kinematic (strain, curvature), mechanical (Young's Modulus, Poisson's Ratio, brittleness), geomechanical (stress, dilation), petrophysical (porosity, permeability), sedimentary (facies proportion), or geophysical (acoustic impedance, p-wave/s-wave velocity). By recognizing that more than one attribute is correlated to and influences the target variable outcomes, predictions of target variables should take into account more than one correlated attribute.
Techniques to solve this multivariate prediction problem generally fall into two categories: Non-parametric and parametric. Non-parametric tools use a subset of the target and attribute data to train complex non-linear relationships that are then applied for predicting the target variable at unsampled locations over the area of interest. Neural networks and variations thereof have become popular for this purpose. Bishop (1995) reviews the fundamentals of setting up and implementing neural networks. Non-parametric techniques are successful in modeling complex multivariate relationships and deriving an optimum combination of the attributes that maximizes the correlation to the target variable. However, the attribute-variable combinations constructed throughout the evolution of the training phase are difficult to control, understand and rationalize with geological interpretation, and merge with prior target variable measurements.
More recently, parametric techniques are being developed and implemented for multivariate prediction. These techniques are generally referred to by the theorem most draw on, Bayesian Updating. Bayesian Updating was first used by Doyen and Den Boer (U.S. Pat. No. 5,539,704) and Doyen et al., 1996. Attributes are combined and merged with prior target variable measurements according to a stationary multivariate Gaussian random function model assumption (Journel and Huijbregts, 1978) and Baye's Law. The result is a posterior Gaussian distribution for the target variable at each prediction location. Unlike for neural network techniques, the attribute-variable combinations are embedded within an objective and understandable mathematical model definition. The most significant challenge with Bayesian Updating is attaining a satisfactory fit to the model parameters in situations with relatively numerous direct target variable measurements.
Guerillot & Roggero (U.S. Pat. No. 5,764,515) use Bayesian inversion with production data to update the geological model of the reservoir. In U.S. Pat. No. 6,950,786, Sonneland and Gehrmann describe a Bayesian and Contextual Bayesian classification method that assigns vectors to a class based on the Gaussian distribution of the classes. Roggero and Mezghani (U.S. Pat. No. 6,662,109, US2003028325) use Bayesian inversion, as a probability density function that predicts observations, accuracy of parameters, and variation in the model. Mickaele and Hu use a pilot point method (U.S. Pat. No. 7,363,163; US2007055447) with kriging to eliminate the need for inversion of the covariance matrix. In U.S. Pat. No. 7,400,978, Mikaele and associates use a gradual deformation technique to model a geological formation over changes in time. Al-Waheed and associates (US2007203681) use a Monte Carlo numerical analysis to account for uncertainties.
Neither a non-parametric or parametric multivariate modeling technique will completely remove the inherent uncertainty in predicting a target variable between measurement locations. Both techniques will however provide means to quantify and manage this uncertainty. For neural network techniques, quantifying uncertainty in predictions is a somewhat non-unique exercise, but is still possible (Wong and Boerner, 2004). For Bayesian Updating the uncertainty is embedded in the posterior Gaussian distribution of the target variable.
As correlated attributes are added, both non-parametric and parametric techniques generally offer increments of diminishing uncertainty. It is the attributes with the highest correlation to the target variable that potentially drive the largest increments towards reducing prediction uncertainty. For neural network applications, the most important attributes are derived from the relative attribute weighting assigned during the training phase (Wong and Boerner, 2004). However, the weighting and resulting relative attribute importance is non-unique since the combination of attributes during training is non-unique.
Unfortunately, as described by Roggero and associates (U.S. Pat. No. 7,392,166) the Bayesian approach is extremely time-consuming forcing them to instead use Gaussian punctual pseudo-data reducing the accuracy of the results. An unbiased model is required that determines with the lowest possible uncertainty a target variable of interest at any location in the model. For practical applications, this must be done in a straightforward theoretically consistent manner. Bayesian Updating (BU) is a statistical theory relating conditional probabilities; the technique underlies all conventional geostatistical algorithms and can be used to achieve multivariate data integration. Its practical application to the geosciences, however, presents a number of challenges.